Improving the Bin Packing Heuristic through Grammatical Evolution Based on Swarm Intelligence

Table 2

Results obtained by each heuristic using (2).

Instance	Fitness function	MTP algorithm	Online heuristics					Offline heuristics
			BestFit	FirstFit	NextFit	WorstFit	Almost WorstFit	BestFit	FirstFit	NextFit	Almost WorstFit	WorstFit
bin1data	Equation (7)	64.307365	316.106050	316.106020	316.106020	316.106000	321.880250	68.090770	68.167305	314.939820	86.119570	76.842740
	Bins used	78378	101097	101097	101097	101097	101705	78660	78661	101097	79314	78843
	Leftover bins	—	22719	22719	22719	22719	23327	282	283	22719	465	936
bin2data	Equation (7)	24.725332	93.025230	93.388504	109.447590	97.613075	115.927666	44.560112	44.561455	110.053760	67.597020	47.655037
	Bins used	20246	23085	23094	23518	23158	23580	20994	20994	23615	21446	21030
	Leftover bins	—	2839	2848	3272	2912	3334	748	748	3369	784	1200
bin3data	Equation (7)	0.390077	1.885825	1.885825	2.586886	2.094760	2.253316	1.390289	1.390289	2.699653	1.518808	1.397016
	Bins used	562	613	613	642	622	631	596	596	650	603	596
	Leftover bins	—	51	51	80	60	69	34	34	88	34	41
binpack1	Equation (7)	0.421557	2.425894	2.604965	7.941016	5.089604	5.365787	0.913949	0.914034	9.504668	1.479180	1.233465
	Bins used	981	1038	1044	1279	1131	1147	995	995	1372	1016	1003
	Leftover bins	—	57	63	298	150	166	14	14	391	22	35
binpack2	Equation (7)	0.180042	2.259154	2.396851	7.989118	4.916397	4.992059	0.705970	0.706004	9.459408	1.043219	0.846457
	Bins used	2032	2154	2162	2669	2342	2353	2062	2062	2851	2087	2068
	Leftover bins	—	122	130	637	310	321	30	30	819	36	55
binpack3	Equation (7)	0.097509	2.014334	2.133326	7.994689	4.725924	4.769628	0.591541	0.591543	9.412333	0.746746	0.666438
	Bins used	4024	4240	4255	5300	4614	4626	4078	4078	5647	4100	4085
	Leftover bins	—	216	231	1276	590	602	54	54	1623	61	76
binpack4	Equation (7)	0.047260	1.838669	1.935710	7.961214	4.597339	4.622161	0.495512	0.495522	9.404397	0.622414	0.572282
	Bins used	8011	8407	8430	10548	9154	9167	8108	8108	11253	8141	8123
	Leftover bins	—	396	419	2537	1143	1156	97	97	3242	112	130
binpack5	Equation (7)	0.000000	0.000000	0.000000	0.000000	0.000000	1.324353	4.830872	4.830872	6.420693	5.453772	4.848756
	Bins used	400	400	400	400	400	420	464	464	491	479	464
	Leftover bins	—	0	0	0	0	20	64	64	91	64	79
binpack6	Equation (7)	0.000000	0.000000	0.000000	0.000000	0.000000	0.682040	4.553619	4.553619	6.197923	4.994128	4.557339
	Bins used	800	800	800	800	800	820	916	916	971	936	916
	Leftover bins	—	0	0	0	0	20	116	116	171	116	136
binpack7	Equation (7)	0.000000	0.000000	0.000000	0.000000	0.000000	0.334901	4.556952	4.556953	6.078939	4.776067	4.558652
	Bins used	1660	1660	1660	1660	1660	1680	1900	1900	2002	1919	1900
	Leftover bins	—	0	0	0	0	20	240	240	342	240	259
binpack8	Equation (7)	0.000000	0.000000	0.000000	0.000000	0.000000	0.167147	4.400591	4.400588	6.059548	4.527734	4.400708
	Bins used	3340	3340	3340	3340	3340	3360	3801	3801	4024	3823	3801
	Leftover bins	—	0	0	0	0	20	461	461	684	461	483
hard28	Equation (7)	0.167276	0.655480	0.655350	13.133352	1.547310	1.927801	0.655480	0.655350	13.133352	1.927801	1.547310
	Bins used	1972	1995	1995	2755	2024	2050	1995	1995	2755	2050	2024
	Leftover bins	—	23	23	783	52	78	23	23	783	52	78
Remaining bins			26423	26484	31602	27936	29133	2163	2164	34322	2447	3508